Proposition 4.20  The restriction \(\chi (u)|_{T_\infty } = (\chi (u_n)|_{T_n})\)  of a quantized character function \(\chi (u)\)  with \(\chi \in K_{-1}^\mathrm {ln}(\vartheta ^t)\)  to the diagonal subgroup \(T_\infty \)  is given by Gorin's \(q^{-2}\) -Schur generating functions \(\mathcal {S}_{q^{-2}}\)  (see [1]}):
\(\chi (u_n)|_{T_n} = \mathcal {S}_{q^{-2}}(t_1,q^2 t_2,\dots ,q^{2(n-1)}t_n;\nu [\chi ]_n),\) 
